The integration constant (http://planetmath.org/SolutionsOfOrdinaryDifferentialEquation) C1 has an influence on the form of the integral curves, but C2 only translates them in the direction of the x-axis.

•

The equation

d2⁢yd⁢x2=f⁢(d⁢yd⁢x)

(6)

is equivalent (http://planetmath.org/Equivalent3) with the normal system

d⁢yd⁢x=z,d⁢zd⁢x=f⁢(z).

(7)

If the equation f⁢(z)=0 has real roots z1,z2,…, these satisfy the latter of the equations (7), and thus, according to the former of them, the differential equation (6) has the solutions y:=z1⁢x+C1, y:=z2⁢x+C2,….

The other solutions of (6) are obtained by separating the variables and integrating: